2021/04/29 by Steven Dale Cutkosky, Cutkosky, Steven Dale, Parangama Sarkar +1 · 2 citations
Computer Science · Mathematics · #13A02 #13A15 #13A18 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2104.14463
openalex publication_date 2021/04/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we define and explore the analytic spread ℓ(\mathcal I) of a filtration in a local ring. We show that, especially for divisorial and symbolic filtrations, some basic properties of the analytic spread of an ideal extend to filtrations, even when the filtration is non Noetherian. We also illustrate some significant differences between the analytic spread of a filtration and the analytic spread of an ideal with examples. In the case of an ideal I, we have the classical bounds ht(I)≤ℓ(I)≤ dim R. The upper bound ℓ(\mathcal I)≤ dim R is true for filtrations \mathcal I, but the lower bound is not true for all filtrations. We show that for the filtration \mathcal I of symbolic powers of a height two prime ideal \mathfrak p in a regular local ring of dimension three (a space curve singularity), so that ht(\mathcal I) =2 and dim R=3, we have that 0≤ ℓ(\mathcal I)≤ 2 and all values of 0,1 and 2 can occur. In the cases of analytic spread 0 and 1 the symbolic algebra is necessarily non-Noetherian. The symbolic algebra is non-Noetherian if and only if ℓ(\mathfrak p(n))=3 for all symbolic powers of \mathfrak p and if and only if ℓ(\mathcal Ia)=3 for all truncations \mathcal Ia of \mathcal I.